In response to your question how I know I am right, the short answer is - I don't - but I beleve I am 'probably right' whereas the ODS is 'probably wrong'. Let's ask ourselves some questions.
A. For Kirman etc not derived from Kirkman
1. Why are a number of places in Lincolnshire KIR... and not KIRK...? For example why not Kirkmington? Because in the Anglo-Saxon derivatives MING meant of the house or following of, as in Burlington. To mix MING and KIRK is highly unlikely.
2. Was the name Kirkman ever prevalent in Lincs? Was it more prevalent elsewhere? If so why would it have migrated in sound and spelling only in Lincs. (see below)
3. Why did 99+% of all Kirman and derivatives live in Lincs. before the Industrial Revolution? (Source: Mormon listings) Why wouldn't the name be transcribed from KIRK to KIR in other places and just occur in Lincs.?
4. KIRK to me sounds like Northern England/Scotland. Was there a significant cultural linkage ever between Lincs and activity north of the Yorkshire? That's what part of the description of KIRK in my Oxford (word) dictionary says ie it relates to Northern England and Scotland. However it also says that it is derived via Old Norse (kirkja) from Old English (cir(i)ce). My mediocre knowlege on Lincs is that it was controlled by the Danes not Norsemen.
5. In a world where religion was so very important, why would someone lose that connection in a name, and the result spread to become so prevalent?
6. Why would the same place that is written as Chevremont in the Doomsday Book and transcribed to Kirmond (Le Mire) not be a reasonable source of the evolved pronunciation and modern spelling? The suggestion doesn't originate with me but an archivist in the Lincoln Archives. This would be a deciding factor- say it was really KIRKMOND ie church on hill and not derived from CHEVREMONT. I have been to Kirmond Le Mire and there is an ancient Anglo Saxon church there - on a promentary but I wasn't struck that it was on some hill. Not that my memory of a transient visit 20 years ago should stand for fact.
B. Re the ODS
1. Is there documented proof of the transition in a family geneology anywhere in the Lincs area where KIRK became KIR? Is it only the writer's opinion or is there something more to go on?
2. How about Kirby or Kirton? It seems reasonable that originally they might have been Kirkby or Kirkton- and if so, there was use of KIRK for church. There exists Kirkstead - which should provide some clues as stead derives from Old English.
So that's what I think - and would be happy to be substantiated or proven wrong - just as long as we have a better understanding.